On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

On the distribution of square-full and cube-full numbers

There are many results on the distribution of square-full and cube-full numbers. In this article the distribution of these numbers are studied in more detail. Suchk-full numbers (k=2,3) are considered which are at the same time 1-free (1k+2). At first an asymptotic result is given for the numberN _k,1(x) ofk-full and 1-free numbers not exceedingx. Then the distribution of these numbers in short intervals is investigated. We obtain different estimations of the differenceN _k,1(x+h)–N_k,1(x) in the casesk=2, 1=4,5,6,7,18 andk=3, 1=5,6,7, 18.     

On the distribution of square-full integers

Let Δ(x) be the error term in the asymptotic formula for the counting function of square-full integers. In the present paper it is proved that , which improves on the exponent 5/33 obtained byX. D. CAO. Project supported by the National Natural Science Foundation of China     

On the distribution of square-full integers

Let A(x) A(x) be the number of square-full integers \leqq x \leqq x and let D(x) \Delta(x) be the error term in the asymptotic formula for A(x) A(x) . Under the Riemann hypothesis, we show that D(x) << x^[12/85]+e \Delta(x)\ll x^{{12\over 85}+\varepsilon} . This improves the earlier results of Zhu and Yu [17], Cao [4, II], Liu [9] and Wu [16], which requires [ 1/7 ] 1\over 7 in place of [ 12/85 ] 12\over 85 .     

The distribution of patterns of inverses modulo a prime

We investigate the distribution of the numbers x∈[1,p] for which all lie in a subset of the set of multiplicative inverses modulo a prime p. Here the a_i are integers coprime to p and the numbers are distinct .     

On the distribution of the orders of 2(mod u) for odd u

In this note we investigate the distribution of odd numbers u such that the order of 2(mod u) is not divisible by qⁿ, where q is an odd prime and Received: 1 July 2004     

On the representation of integers as sums of triangular numbers

Summary In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. Ifn 0 is a non-negative integer, then thenth triangular number isT _n =n(n + 1)/2. Letk be a positive integer. We denote by _k (n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculate _k (n). The case wherek = 24 is particularly interesting. It turns out that, ifn 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 2¹²(2¹² – 1) ₂₄(n – 3). Furthermore the formula for ₂₄(n) involves the Ramanujan(n)-function. As a consequence, we get elementary congruences for(n). In a similar vein, whenp is a prime, we demonstrate ₂₄(p ^k – 3) as a Dirichlet convolution of ₁₁(n) and(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres.     

Distribution of exponential functions with squarefull exponent in residue rings

For a real x ≥ 1 we denote by S[x] the set of squarefull integers n ≤x, that is, the set of positive integers n ≤ such that l²|n for any prime divisor l|n. We estimate exponential sums of the form

     

Evaluation of the Dedekind zeta functions of some non-normal totally real cubic fields at negative odd integers

Let {K _m } _{m ≥ 4} be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial f _m (x) = x ³ − mx ² − (m + 1)x − 1, where m is an integer with m ≥ 4. In this paper, we will apply Siegel’s formula for the values of the zeta function of a totally real algebraic number field at negative odd integers to K _m , and compute the values of the Dedekind zeta function of K _m . This work was supported by grant No.R01-2006-000-11176-0 from the Basic Research Program of KOSEF.     

Lower bound of overall exponential sums for polynomialsϕ(x d )

LetS ⁽¹⁾ (n, Q) denote the maximum module of exponential sums for polynomials of degree n over the Galois fieldF _Q . In a previous paper the transition to the multiple exponential sums allowed us to obtain a good lower bound of the valueS ⁽¹⁾ (n, Q), which coincides with Weil's bound whenn = q ^(m-1)/2 + 1, whereq, m are odd andm 3. Here the same approach is used for the estimation of the valueS ^(d) (n, Q), which corresponds to polynomials(x ^d ) overF _Q , whered is any divisor ofq – 1.     

On the distribution of irreducible algebraic integers

We study large values of the remainder term in the asymptotic formula for the number of irreducible integers in an algebraic number field K. In the case when the class number h of K is larger than 1, under certain technical condition on multiplicities of non-trivial zeros of Hecke L-functions, we detect oscillations larger than what one could expect on the basis of the classical Littlewood’s omega estimate for the remainder term in the prime number formula. In some cases the main result is unconditional. It is proved that this is always the case when h = 2. Author’s address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland The author was supported in part by KBN Grant # N N201 1482 33.     

Primes in quadratic progressions on average

In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.     

The asymptotic behaviour of symbolic generic initial systems of generic points

Consider the ideal I⊆K[x,y,z]$I\subseteq K[x,y,z]$ corresponding to points _p1,…,_pr$p_1,\dots ,p_r$ of P²${\mathbb{P}}^2$. We study the symbolic generic initial system  _{gin_(I(m))}m${\left\{\text{gin}\left(I^{\left(m\right)}\right)\right\}}_m$ of such an ideal and its behaviour as m$m$ gets large. In particular, we describe the limiting shape   of this system explicitly when _p1,…,_pr$p_1,\dots ,p_r$ are generic, assuming that the Uniform SHGH Conjecture holds for r≥9$r\ge 9$. The symbolic generic initial system and its limiting shape reflect information about the asymptotic behaviour of Hilbert functions of fat point ideals.     

INJECTIVE RESOLVENTS AND PREENVELOPES

Abstract

Let R be a noetherian ring, and denote the full subcategories of R-modules L such that Extⁱ(E,L)=0 for all injective R-modules E for 1?i?n and O?i?n by C_n, and C′_n respectively. Then LεC_n, if and only if every injective resolution of L is an injective resolvent of the nth cosyzygy. In this case, L is not injective if and only if its injective dimension is greater than n. If LεC′_n and idN?n. then Hom(N,L)=0 for all R-modules N. As an application, let K_n be the nth syzygy of an injective resolvent of the nth cosyzygy of an R-module N, then there exists a homomorphism φ:N → K such that ((φ,i_N), K_n ? E(N)) and (φ,K_n) are preenvelopes of N for C_s and C′_s respectively, for s≥n. If the global dimension of R is at most 2, then C′₁ is reflective in the category of R-modules.     

Irreducible algebraic integers in short intervals

We study distribution of irreducible algebraic integers in short intervals and prove that if the class number of an algebraic number field K exceeds 2, every interval of the form (x, x + x ^θ) with a fixed θ > 1/2 contains absolute value of the norm of an irreducible algebraic integer from K provided x ≥ x ₀. The constant x ₀ effectively depends on K and θ. The author was supported in part by the grant no. N N201 1482 33 from the Polish Ministry of Science and Higher Education.